# Difference between revisions of "Compton Catastrophe"

(Created page with '===Short Topical Videos=== * ===Reference Material=== * <latex> \documentclass[11pt]{article} \def\inv#1{{1 \over #1}} \def\ddt{{d \over dt}} \def\mean#1{\left\langle {#1}\ri…') |
|||

Line 1: | Line 1: | ||

===Short Topical Videos=== | ===Short Topical Videos=== | ||

− | * | + | * [https://youtu.be/HeuuX31Cyq0 Synchrotron Self-Interactions (Aaron Parsons)] |

===Reference Material=== | ===Reference Material=== |

## Latest revision as of 00:17, 3 November 2015

### Short Topical Videos[edit]

### Reference Material[edit]

<latex> \documentclass[11pt]{article} \def\inv#1Template:1 \over \def\ddtTemplate:D \over dt \def\mean#1{\left\langle {#1}\right\rangle} \def\sigot{\sigma_{12}} \def\sigto{\sigma_{21}} \def\eval#1{\big|_{#1}} \def\tr{\nabla} \def\dce{\vec\tr\times\vec E} \def\dcb{\vec\tr\times\vec B} \def\wz{\omega_0} \def\ef{\vec E} \def\ato{{A_{21}}} \def\bto{{B_{21}}} \def\bot{{B_{12}}} \def\bfieldTemplate:\vec B \def\apTemplate:A^\prime \def\xp{{x^{\prime}}} \def\yp{{y^{\prime}}} \def\zp{{z^{\prime}}} \def\tp{{t^{\prime}}} \def\upxTemplate:U x^\prime \def\upyTemplate:U y^\prime \def\e#1{\cdot10^{#1}} \def\hf{\frac12} \def\^{\hat } \def\.{\dot } \def\tnTemplate:\tilde\nu

\usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document} \def\numin{{\nu_{min}}} \def\numax{{\nu_{max}}} \def\gamin{\gamma_{min}} \def\gamax{\gamma_{max}} \def\gul{{min(\sqrt{\nu\over\numin},\gamax)}} \def\gll{{max(\sqrt{\nu\over\numax},\gamin)}} \subsection*{ Compton Catastrophe}

If you keep scattering the same electrons, as in Synchrotron Self-Compton,
there is a danger, if things are dense enough, of a runaway amplification
of radiation energy density, or a ``Compton Cooling Catastrophe*. However,*
we've never seen anything with a brightness temperature of $10^{12}K$.
What sets this ``inverse Compton limit* at this temperature? Comparing,*
for a single electron,
the luminosity of inverse Compton scattering to synchrotron scattering:
$${L_{IC}\over L_{sync}}={{4\over3}\beta^2\gamma^2\sigma_TcU_{ph}\over
{4\over3}\beta^2\gamma^2\sigma_TcU_B}={U_{ph}\over U_B}
\begin{cases} >1&catastrophe\\ <1 &no\ catastrophe\end{cases}$$
Now we're going to make an approximation that we are on the Rayleigh-Jeans
side of the blackbody curve, so that:
$$\begin{aligned}U_{ph}=U_{ph,sync}&\propto\nu_mI_\nu(\nu_m)\\
&\propto\nu_m{2kT_B\over\lambda_m^2}\\
&\propto\nu_m^3T_B\\ \end{aligned}$$
where $\nu_m$ is the frequency of peak of synchrotron emission.
Now $U_B\propto B^2$ is pretty obvious:
$$\nu_m\sim\gamma_m^2\nu_{cyc}\propto\gamma_m^2B$$
where this $\gamma_m$ is not $\gamax$. Making the approximation that we
are in the optically thick synchrotron spectrum, so that $\gamma m_ec^2\sim
kT$, then we get $\nu_m\sim T_B^2B$. We can say that the kinetic temperature
is the brightness temperature because we are talking about the average kinetic
energy of the electrons generating the synchrotron radiation with a particular
brightness temperature (i.e. another frequency of synchrotron radiation will
have another brightness temperature, and another set of electrons moving
with a different amount of kinetic energy). Thus,
$${U_{ph}\over U_B}=C{\nu_m^3T_B\over\nu_m^2}T_B^4
=\left({\nu_m\over10^9Hz}\right)\left({T_B\over10^{12}K}\right)^5=1$$
A way of think about this is that, in order to avoid having infinite energy
in this gas of electrons, there has to be a limit on the brightness
temperature (which is determined by the density of electrons). This is a
self-regulating process--if the brightness temperature goes too high, an
infinite energy demand is set up, knocking it back down.

\end{document} <\latex>